College

13. Find \((f \cdot g)(x)\) if \[f(x) = 7x^3 - 5x^2 + 42x - 30\] and \[g(x) = 7x - 5\].

A. \((f \cdot g)(x) = 49x^4 + 70x^3 - 319x^2 + 420x - 150\)

B. \((f \cdot g)(x) = 49x^4 - 269x^2 - 150\)

C. \((f \cdot g)(x) = 49x^4 + 269x^2 + 150\)

D. \((f \cdot g)(x) = 49x^4 - 70x^3 + 319x^2 - 420x + 150\)

Answer :

To find [tex]\((f \cdot g)(x)\)[/tex], we need to multiply the two polynomials given, [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex].

Given:
[tex]\[ f(x) = 7x^3 - 5x^2 + 42x - 30 \][/tex]
[tex]\[ g(x) = 7x - 5 \][/tex]

Now, we'll perform polynomial multiplication between these two expressions:

1. Multiply each term of [tex]\(g(x)\)[/tex] by each term of [tex]\(f(x)\)[/tex]:

- Start with the highest degree term from [tex]\(g(x)\)[/tex], which is [tex]\(7x\)[/tex]:
- [tex]\(7x \cdot 7x^3 = 49x^4\)[/tex]
- [tex]\(7x \cdot (-5x^2) = -35x^3\)[/tex]
- [tex]\(7x \cdot 42x = 294x^2\)[/tex]
- [tex]\(7x \cdot (-30) = -210x\)[/tex]

2. Now multiply the constant term [tex]\(-5\)[/tex] from [tex]\(g(x)\)[/tex] by each term of [tex]\(f(x)\)[/tex]:

- [tex]\(-5 \cdot 7x^3 = -35x^3\)[/tex]
- [tex]\(-5 \cdot (-5x^2) = 25x^2\)[/tex]
- [tex]\(-5 \cdot 42x = -210x\)[/tex]
- [tex]\(-5 \cdot (-30) = 150\)[/tex]

3. Combine all the terms from the distributed multiplication:

- [tex]\(49x^4\)[/tex]
- [tex]\(-35x^3 + (-35x^3) = -70x^3\)[/tex]
- [tex]\(294x^2 + 25x^2 = 319x^2\)[/tex]
- [tex]\(-210x + (-210x) = -420x\)[/tex]
- [tex]\(150\)[/tex]

4. Write the resulting polynomial expression:

[tex]\[
(f \cdot g)(x) = 49x^4 - 70x^3 + 319x^2 - 420x + 150
\][/tex]

Therefore, the product of the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex], [tex]\((f \cdot g)(x)\)[/tex], is [tex]\[49x^4 - 70x^3 + 319x^2 - 420x + 150\][/tex]. This matches the correct option provided earlier.

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